Problem: Let N be a positive odd number.

There are N coins, numbered 1, 2, \ldots, N. For each i (1 \leq i \leq N), when Coin i is tossed, it comes up heads with probability p_i and tails with probability 1 - p_i.

Taro has tossed all the N coins. Find the probability of having more heads than tails.

Constraints

* N is an odd number.
* 1 \leq N \leq 2999
* p_i is a real number and has two decimal places.
* 0 < p_i < 1

Input

Input is given from Standard Input in the following format:


N
p_1 p_2 \ldots p_N


Output

Print the probability of having more heads than tails. The output is considered correct when the absolute error is not greater than 10^{-9}.

Examples

Input

3
0.30 0.60 0.80


Output

0.612


Input

1
0.50


Output

0.5


Input

5
0.42 0.01 0.42 0.99 0.42


Output

0.3821815872

Solution (in CPP):

Solution:
#include<bits/stdc++.h>   //https://atcoder.jp/contests/dp/tasks/dp_i
using namespace std;
double dp[3000][3000],arr[3000],sum;
int n;
int main()
{
	cin>>n;
	dp[0][0]=1;
	for(int i=1;i<=n;++i){cin>>arr[i];dp[i][0]=dp[i-1][0]*arr[i];}
	int t=(n-1)/2;
	for(int i=1;i<=n;++i)
	{
		for(int j=1;j<=i&&j<=t;++j)
		{
			dp[i][j]=dp[i-1][j-1]*(1-arr[i])+dp[i-1][j]*arr[i];
		}
	}
	for(int i=0;i<=t;++i)sum+=dp[n][i];
	printf("%.10lf",sum);
}